Treating stabilizer operations as free, we establish lower bounds on the number of resource states, also known as magic states, needed to perform various quantum computing tasks. Our bounds apply to adaptive computations using measurements with an arbitrary number of stabilizer ancillas. We consider (1) resource state conversion, (2) single-qubit unitary synthesis, and (3) computational subroutines including the quantum adder and the multiply-controlled Z gate. To prove our resource conversion bounds we introduce two new monotones, the stabilizer nullity and the dyadic monotone, and make use of the already-known stabilizer extent. We consider conversions that borrow resource states, known as catalyst states, and return them at the end of the algorithm. We show that catalysis is necessary for many conversions and introduce new catalytic conversions, some of which are optimal. By finding a canonical form for post-selected stabilizer computations, we show that approximating ...

中文翻译：

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非Clifford资源的下界用于量子计算

将稳定器操作视为自由操作，我们为执行各种量子计算任务所需的资源状态（也称为魔术状态）的数量设置了下限。我们的界限适用于使用具有任意数量的稳定器微镜的测量值进行的自适应计算。我们考虑（1）资源状态转换，（2）单量子位unit合成，以及（3）包括量子加法器和乘法控制Z门的计算子例程。为了证明我们的资源转换范围，我们引入了两个新的单调，稳定器无效性和二元单调，并利用了已知的稳定器范围。我们考虑借用资源状态（称为催化剂状态）的转换，并在算法结束时将其返回。我们证明了催化对于许多转化都是必要的，并引入了新的催化转化，其中一些是最佳的。通过找到用于后选稳定器计算的规范形式，我们证明了近似...